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Use transformations of the graph of \( y=1 / x \) to graph the rational function, as in Erample 2 . \[ s(x)=\frac{3}{x+1} \]

Ask by Todd Barker. in the United States
Mar 12,2025

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To graph \( s(x) = \frac{3}{x+1} \), start with the parent function \( y = \frac{1}{x} \). Shift it left by 1 unit and stretch it vertically by 3. Draw the vertical asymptote at \( x = -1 \) and horizontal asymptote at \( y = 0 \). Plot key points like \( (-2, -3) \), \( (0, 3) \), and \( (1, 1.5) \), and sketch the curve accordingly.

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To graph the function \( s(x) = \frac{3}{x+1} \), start by recognizing its relation to the parent function \( y = \frac{1}{x} \). The transformation involved shifts the graph left by 1 unit, as indicated by the \( x + 1 \) in the denominator. Next, to account for the vertical stretch, observe that the numerator has a 3, which will make the graph reach three times higher than the basic \( \frac{1}{x} \). Next, plot the vertical asymptote at \( x = -1 \) and the horizontal asymptote at \( y = 0 \). To help visualize the points, consider evaluating s(x) at key points: when \( x = -2 \), \( s(-2) = -3 \) and when \( x = 0 \), \( s(0) = 3 \). These points will guide you in shaping the curve correctly, reflecting the rational function's characteristic behavior near the asymptotes!

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